`Evaluate int a^logxdx, a gt 0 ,x gt 0`

Evaluate: int(1+x)logxdx

Evaluate: int(x^2+1)logxdx

Let l= int_(0)^(pi//2)(cosx)/(a cos x+ b sinx)dx and J=int_(0)^(pi//2)(sin x)/(a cosx+b sin x )dx , where a gt 0 and b gt 0 Compute the values of l and j

If I=int_(-pi)^(pi) (sin^(2))/(1+a^(x))dx, a gt 0 , then I equals

Evaluate Lim_(x to 0) (log (a+x) - log(a-x))/x , a gt 0

The value of the constant a gt 0 such that int_(0)^(a) [tan^(-1)sqrt(x)]dx=int_(0)^(a) [cot^(-1)sqrt(x)]dx , where[.] denotes the greatest integer function, is

Evaluate the limits, if exist lim_(x-gt0)(log_e(1+2x))/(x)

Find the maximum value of the function (log x)/x " when "x gt 0 .

The least value of the function f(x) = ax + (b)/(x) (x gt 0, a gt 0, b gt 0)

Find the number of solutions to cos x = (x)/(10), x gt 0.